The argument is a minor modification of [1]}. For simplicity, let us outline the proof in the case when \(U\)  is abelian and acts with zero dimensional stabilisers on \(Z^{ss}_{\min }\) ; if \(U\)  is not abelian, one needs to successively take quotients by the abelian groups appearing as quotients in a normal series and, if the stabilisers are positive dimensional, one needs to choose complementary subgroups as in the proof of Theorem REF .
First one constructs a geometric \(U\) -quotient of \(Y^{ss}_{\min }\)  similarly to the affine local construction of [1]}; however, one instead takes a basis \(\lbrace \sigma _i\rbrace _{i=1}^r\)  of the \(\hspace{0.83328pt}\overline{\hspace{-0.83328pt}L\hspace{-0.83328pt}}\hspace{0.83328pt}\) -invariant sections of maximal \(\lambda (\mathbb {G}_m)\) -weight \(H^0(Y,\mathcal {L})^{\hspace{0.83328pt}\overline{\hspace{-0.83328pt}L\hspace{-0.83328pt}}\hspace{0.83328pt}}_{\max }\)  (recall that these sections are also \(U\) -invariant), then \(Y_{\min }^{ss}\)  is covered by the open affine sets \(Y_{\sigma _i}\)  and each of these admits a trivial geometric \(U\) -quotient given by taking rings of \(U\) -invariants (cf. [1]}). Since these invariant rings are finitely generated, one can pick \(s\)  so they are all generated in degree \(s\) . Let \(W:=(H^0(Y,\mathcal {L}^{\otimes s})^U)^*\) ; then we have a natural \(U\) -invariant morphism \(\phi : Y^{ss}_{\min } \rightarrow \mathbb {P}:=\mathbb {P}(W)\) , which thus factors via the quotient \(q_U : Y^{ss}_{\min } \rightarrow Y^{ss}_{\min }/U\)  and a locally closed immersion \(\overline{\phi } :Y^{ss}_{\min }/U \rightarrow \mathbb {P}\) . Moreover, there is an induced \(L\) -action on \(\mathbb {P}\)  such that \(\phi \)  is \(L\) -equivariant; we denote the corresponding (semistable) \(\lambda \) -minimal weight space \(Z^{(ss)}_{\min }(\mathbb {P})\)  and the corresponding open by \(\mathbb {P}_{\min }^{(ss)}\) . After appropriately twisting the linearisation on \(\mathbb {P}\)  so it is well-adapted, we have \(\mathbb {P}^{\lambda -(s)s} = \mathbb {P}_{\min } \setminus Z_{\min }(\mathbb {P})\)  by the Hilbert–Mumford criterion.
The same argument as in [1]} shows that the image of \(\overline{\phi }\)  is contained in \(\mathbb {P}_{\min }^{ss}\)  as a closed subscheme. Moreover, for \(y \in Y_{\min }^{ss}\) , one has \(\phi (y) \in Z^{ss}_{\min }(\mathbb {P})\)  if and only if \(y \in UZ_{\min }^{ss}\)  analogously to [1]}. Consequently, \(\phi \)  induces a closed immersion
\( (Y_{\min }^{ss} \setminus U Z_{\min }^{ss})/U \hookrightarrow \mathbb {P}_{\min }^{ss} \setminus Z_{\min }^{ss} = \mathbb {P}_{\min }^{ss} \times _{\mathbb {P}} \mathbb {P}^{\lambda -s}.\) 
Thus one obtains a geometric quotient \((Y^{ss}_{\min }\setminus UZ_{\min }^{ss})/\widehat{U}\)  as a locally closed subscheme of the projective scheme \(\mathbb {P}/\!/ \lambda (\mathbb {G}_m)\) ; in particular, this \(\widehat{U}\) -quotient is not necessarily projective.
The reductive GIT quotient \(q_L: \mathbb {P}\dashrightarrow \mathbb {P}/\!/L\)  factors via \(\mathbb {P}/\!/ \lambda (\mathbb {G}_m)\)  and we will construct our \(H\) -quotient as a closed subscheme of \(\mathbb {P}/\!/L\) . A reductive Hilbert–Mumford argument shows \(\mathbb {P}^{L-ss} \subset \mathbb {P}^{ss}_{\min } \setminus Z^{ss}_{\min }\) , and since \((Y_{\min }^{ss} \setminus U Z_{\min }^{ss})/U\)  is closed in the latter, the inclusion
\(\mathbb {P}^{L-ss} \times _{\mathbb {P}} (Y_{\min }^{ss} \setminus U Z_{\min }^{ss})/U \hookrightarrow \mathbb {P}^{L-ss} \) 
is closed and \(Y/\!/H:=q_L( \mathbb {P}^{L-ss} \times _{\mathbb {P}} (Y_{\min }^{ss} \setminus U Z_{\min }^{ss})/U)\)  is a closed subscheme of \(\mathbb {P}/\!/L\) . The composition \(Y_{\min }^{ss} \setminus U Z_{\min }^{ss} \rightarrow Y/\!/H\)  is thus a projective good \(H\) -quotient.
If in addition, we assume that REF  holds, then \(\operatorname{dom}(q) = Y^{H-s}\)  as in Theorem REF . To conclude that \(Y/\!/H\)  is given by taking the projective spectrum of the ring of invariants for a well-adapted perturbation of the linearisation, it suffices to show this quotient coincides with the enveloping quotient by [6]}. To prove this, one first shows that \(\operatorname{dom}(q)\)  is contained in the locally trivial stable locus with respect to \(H\)  (see [6]}) and then deduces that \(Y/\!/H\)  coincides with the quotient of the locally trivial stable locus and the enveloping quotient as in [1]}.
